PHYSICS OF PLASMAS
VOLUME 7, NUMBER 5
MAY 2000
Growth and nonlinear evolution of the modified Simon-Hoh instability in an electron beam-produced plasma* Y. Sakawa†,a) and C. Joshib) Electrical Engineering Department, University of California, Los Angeles, California 90095
共Received 17 November 1999; accepted 2 February 2000兲 The growth and nonlinear evolution of the modified Simon–Hoh instability 共MSHI兲 is observed in a weak electron beam-produced collisionless cylindrical plasma, in which electrons are strongly magnetized and the ions are essentially unmagnetized. The evolution of this instability occurs through a sequence of sideband instabilities, thought to be induced by trapped ions, and a period doubling sequence. Transient study of the MSHI reveals that the growth rate of the MSHI is extremely rapid; of the order the instability frequency. © 2000 American Institute of Physics. 关S1070-664X共00兲95205-4兴
thought to arise because of ion trapping effects.5,6 Each transition begins with the appearance of an apparently new oscillation mode at a very low frequency. The frequency rises to some subharmonic value of the fundamental frequency and stays locked with it. This is followed by the appearance of another low-frequency oscillation mode whose frequency rises to typically half the value of the previous lowest frequency where it meets and locks with the decreasingfrequency, local lower-sideband frequency. Such sequence of successive sideband instabilities and locking may be common to plasma systems in which large ion orbit dynamics are important and applicable to many systems involving excitation of coherent modes. We also find that the nonlinear evolution of the MSHI can occur not only via excitation of a sequence of sideband instability 共modulational instabilities兲 but also via a period doubling sequence.7 This paper is organized as follows: Part II is a review of MSHI; in Part III transient study and the steady-state characteristics of MSHI are presented; Part IV discusses various scenarios for the evolution of MSHI the conclusion and a summary of the paper are given in Part V.
I. INTRODUCTION
The transition from a simple unstable equilibrium dominated by linearized instabilities to a turbulent state in the plasma is generally very complex. This is because plasmas have a rich variety of collective modes of oscillation and a great number of nonlinear coupling mechanisms. Furthermore, in plasma experiments one is typically unable to control the plasma parameters with enough precision so that subtle phenomena near the transition to turbulence may be studied. Our aim is to identify a model plasma system and study the evolution of such a plasma from one coherent state into turbulence. The plasma system chosen for this study is a low-density electron beam-produced and magnetically confined plasma column, which is extremely low in noise. In the regime where the neutral gas pressure is low and the electron beam current is small, such a plasma column is seen to be unstable to one mode, which we identify as the modified Simon–Hoh instability 共MSHI兲,1 which has an instability mechanism similar to the collisional Simon–Hoh instability.2–4 This instability is seen to occur in a cylindrical collisionless plasma if a radial dc 共direct current兲 electric field and a radial density gradient both exist in the same direction. In Ref. 1 the steady-state characteristics of the MSHI are described in detail while in Ref. 5, one route taken by the nonlinear evolution of the MSHI was described. In this paper, we review the previous work and in addition, we document the growth phase and the steady-state characteristics of the MSHI. Theory shows and experiments confirm, that the growth rate of the MSHI is on the order of the instability frequency; i.e., the instability is in the strongly driven regime. We then document the nonlinear evolution of this mode. One specific process which leads to an eventual turbulent spectrum in this plasma is the evolution of the MSHI through a series of sideband instabilities that are
II. REVIEW OF MSHI
Simon,2 Hoh,3 and Thomassen4 studied the Simon–Hoh instability 共SHI兲 in a weakly ionized, inhomogeneous, collisional, magnetized plasma under a strong radial electric field E r0 perpendicular to the dc axial magnetic field B 0 . The SHI is unstable when the density gradient ⵜn 0 and E r0 are in the same direction. Due to collisions with neutral particles, ion E r0 ⫻B 0 drift velocity is slower than that for electrons. The difference between the electron and ion E⫻B drift velocities causes a space charge separation between the electron and ion density perturbations in the direction, and consequently produces a perturbed azimuthal electric field, E 1 . When the plasma density is inhomogeneous and ⵜn 0 •Er0 ⬎0, then the E 1 ⫻B 0 velocity enhances the density perturbation. In collisionless, weakly magnetized-ion plasmas 共where electrons are magnetized兲, a similar instability, which we call the modified Simon–Hoh instability 共MSHI兲,1 occurs due to the slower 共relative to the electrons兲 ion drift velocity caused by
*Paper HI24 Bull. Am. Phys. Soc. 44, 157 共1999兲. † Invited speaker. a兲 Also at: Department of Energy Engineering and Science, Nagoya University, Nagoya 464-8603, Japan. b兲 Electronic mail:
[email protected] 1070-664X/2000/7(5)/1774/7/$17.00
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© 2000 American Institute of Physics
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Phys. Plasmas, Vol. 7, No. 5, May 2000
Growth and nonlinear evolution of the modified . . .
the large ion Larmor radius effect. In our experiment, the ions are unmagnetized because of the small B 0 and large E r0 . The smaller ion azimuthal drift velocity compared with the electron E r0 ⫻B 0 velocity causes a space charge separation between the electron and ion density perturbations in the direction. The consequent perturbed azimuthal electric field E 1 and the enhancement of the density perturbation by the E 1 ⫻B 0 velocity are the same as in the SHI case. Using a fluid theory, the portion of the density perturbations of the magnetized electron-fluid ˜n e /n 0 and unmagnetized ion-fluid ˜n i /n 0 共with azimuthal Doppler shift and neglecting radial variations兲 in the frequency f range of f ci ⬍ f ⬍ f ce ( f ci and f ce are the ion and electron cyclotron frequencies, respectively兲 are given by ˜n e n0 ˜n i n0
⫽ ⫽
e
*
Te ⫺E e
共1兲
,
c s2 k 2
T e 共 ⫺k v i 兲 2
共2兲
,
where e is the electron charge, is the fluctuating potential, T e is the plasma electron temperature, * ⫽⫺k (T e /eB 0 ) ⫻(1/n 0 )(dn 0 /dr) is the electron diamagnetic drift frequency, E ⫽⫺k E r0 /B 0 is the electron E⫻B drift frequency, k is wave number in direction, c s ⫽ 冑T e /M , M is ion mass, and v i is the mean azimuthal ion drift speed. Typical experimental parameters and the derivation of Eqs. 共1兲 and 共2兲 are summarized elsewhere.1 Assuming that ˜n i ˜ e , the dispersion relation is given by ⫽n
* ⫺E
⫽
k 2 c s2 共 ⫺k v i 兲 2
共3兲
.
Equation 共3兲 is solved to give
R ⫽k v i ⫹
I⫽
⯝
冑 冑
c s2 k 2 2*
⯝k v i ,
c s2 k 2 共 E ⫺k v i 兲
* c s2 k 2 共 E ⫺k v i 兲
*
⫺
共4兲 c s4 k 4 4 *2
,
共5兲
where R and I are the real and the imaginary parts of the instability frequency, respectively. This shows that when v i ⬍E r0 /B 0 and E / * ⬎0, we can have an excitation of a fluid instability. Note that the perpendicular phase velocity is nearly v i with a small correction due to the second term in Eq. 共4兲. III. TRANSIENT STUDY AND STEADY-STATE CHARACTERISTICS OF MSHI
The experimental setup is shown elsewhere.1 A 1 cm 关full width at half maximum 共FWHM兲兴 diameter Gaussian electron beam was injected axially into one end of a 10 cmdiam, 180 cm long stainless steel vacuum vessel, immersed in a dc magnetic field B⫽50– 320 G. Most of our measure-
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ments were conducted at B⫽160 G. Ar gas was used for most of the measurements, which were carried out over the pressure range P⫽5⫻10⫺6 ⫺5⫻10⫺5 Torr 共the vacuum base pressure ⯝5⫻10⫺7 Torr). The ion mass dependence of the observed instability was determined by using Xe, Kr, and N2 in addition to Ar. Electrons were emitted from a directly heated spiral tungsten filament cathode, accelerated through a 1 cm diam hole of the grounded anode, which was biased positive with respect to the cathode, and collected by the grounded endplate target 共1.2 cm diam兲. The distance between the electron gun and the endplate target defined the plasma length L p to be 80 cm. The electron current I b was measured at the endplate target, and the experiments were performed typically at the acceleration voltage of V B ⫽250 V and I b ⫽10– 1000 A, which was governed by temperature limited emission. At these low values of pressure and current, the system was always below the beam-plasma discharge8 threshold and thus could be kept remarkably free of noise 共up to ⫺70 dBV below the amplitude of the fundamental mode兲. In the steady state, densities of beam electrons n b , plasma electrons n e , plasma ions n i and T e were measured by the Langmuir probe. As a result, n b ⯝104 ⫺107 cm⫺3 , n i ⯝105 ⫺108 cm⫺3 , and T e ⯝4 eV were obtained at the plasma center. In contrast to the nearly 1 cm 共FWHM兲 Gaussian profiles of n e and n b , n i showed a broader profile from the beam center to the chamber wall. The difference between n e and n i profiles implies the existence of a radial dc electric field, E r0 . We deduced E r0 by measuring the radial profile of the plasma potential ⌽ 关Fig. 1共a兲兴 with an emissive probe.1 Whereas T e ⯝4 eV, the ion perpendicular temperature T i⬜ measured by an energy analyzer showed good agreement with the depth of the dc potential ⌬⌽.1 This is suggestive of T i⬜ as being the characteristic kinetic energy of the ions as they rattle back and forth in the radial potential well. In this experiment (B⫽160 G, Ar plasma兲 the electrons are strongly magnetized (r Le ⫽0.04 cm for T e ⫽4 eV) whereas the ions are essentially unmagnetized (r Li ⯝5.6 ⫺17.7 cm for T i⬜ ⯝1⫺10 eV), where r Le and r Li are the electron and ion Larmor radius, respectively. Most of the instability measurements were made with unbiased, grounded cylindrical Langmuir probes, which were terminated by a 1 k⍀ resistor, in order to collect electron current with minimal perturbation on plasma. However, the same results, with much reduced amplitudes, were obtained with probes biased to give the ion saturation current. The frequency spectra were obtained both by performing fast Fourier transforms 共FFT兲 on the real-time signal and with an HP model 3561A spectrum analyzer. At I b less than few A, we first observe an instability M 1 mode, whose frequency f 1 ranges in f ci ⬍ f 1 ⬍ f ce . We identified this mode as the MSHI.1 It is an m⫽1 azimuthal mode and n 1 peaks at r⯝3 to 4 mm from the beam axis 关Fig. 1共b兲兴.1 The measured value of f 1 is consistent with a calculated effective ion E⫻B drift frequency f Ei , which takes into account the large ion Larmor radius effect,1 and the the azimuthal ion drift frequency f i ⫽ v i /(2 r), where v i is the measured value of the mean azimuthal velocity of the
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Phys. Plasmas, Vol. 7, No. 5, May 2000
Y. Sakawa and C. Joshi
FIG. 2. Radial position dependence of the electron saturation current measured at 共a兲 r⫽0 cm and 共b兲 r⫽1.0 cm. Vertical scale in 共b兲 is expanded ten times more than in 共a兲. Ar 2⫻10⫺5 Torr and I b ⫽2000 A. 共c兲 Real 共solid lines兲 and imaginary 共dashed lines兲 parts of the dispersion relation of the MHSI. ⍀ i ⫽2 f ci and k v E ⫽ E . At a typical experimental condition 共Ar 2⫻10⫺5 Torr and I b ⫽100 A), k v E /⍀ i ⫽155 共Ref. 1兲.
FIG. 1. 共a兲 Radial profiles of dc plasma potential ⌽ at P⫽2⫻10⫺5 Torr. 共b兲 Radial profiles of the log of spectral amplitude 共in dBV兲 of M 1 , M 2 , and M s modes. 0 dbV⫽1 V. I b ⫽100 A and P⫽1⫻10⫺5 Torr. 共c兲 I b dependence of f 1 , f i , and f Ei . Here, f i ⫽ v i /(2 r 1 ) and r 1 ⫽0.5 cm is the radius at which the spectral amplitude of the M 1 mode (n 1 ) is maximum. 共d兲 The temporal evolution of the ion radial density profiles measured at high-pressure Ar gas ( P⫽3⫻10⫺4 Torr) and at high-beam current (I b ⫽3 mA). Steady-state density profile of the plasma electron n e is also shown. P⫽2⫻10⫺5 Torr and I b ⫽500 A. 共e兲 Ion mass A i 关 H2 (A i ⫽2), Ar, and Xe at P⫽10⫺3 Torr] and Ar pressure P dependence of the diffusion velocity v Di . 共f兲 Dependencies of f 1 , f 2 , and f b on I b at P⫽2 ⫻10⫺5 Torr. Values of f b calculated from 1 measured at two different radii are shown.
ions 共using a one-sided probe9兲, over a wide range of I b 关Fig. 1共c兲兴.1,5 For the MSHI to occur there has to be a radial electric field E r0 . This in turn arises because the ions have a wider radial profile than the electrons. To understand how the ions establish this profile, we carry out ion diffusion measurements with a pulsed electron beam by applying a pulsed V B . The rise time of I b is 0.4 s. However, the rise time of V B is much less than 0.4 s. Figure 1共d兲 shows the temporal evolution of the ion radial density profiles reconstructed from the time variation of the ion saturation current I is measured at seven different radial positions, r⫽0, 0.5, 1.0, 2.0, 3.0, 4.0, and 5.0 cm from the beam center. The n i profile is the same as the beam profile early in time. Then ions start diffusing, and at t ⫽30– 100 s they have a broad profile observed in the steady state. Figure 1共e兲 shows the ion mass A i and the Ar pressure P dependence of the ion diffusion velocity v Di . We see that v Di is larger for the smaller A i and the lower P.
In order to explain the measured results, we first considered ambipolar ion diffusion. However, the observed ion diffusion velocity v Di for P⯝10⫺5 ⫺10⫺3 Torr shown in Fig. 1共d兲 is much larger than the calculated ambipolar ion diffusion velocity 共when T i ⯝T e ) v⬜A ⯝23⫺2.3⫻103 cm/s. It is pointed out by Simon10 that short-circuit effect is important in short plasma columns with the magnetic-field lines terminated by conducting plates.11 In our experiments, the condition for the perpendicular diffusion to be nonambipolar12 is satisfied and we expect to have the short-circuit effect and, therefore, the nonambipolar radial diffusion. The calculated ion diffusion velocity v⬜i versus the neutral pressure P for Xe, Ar, and H2 with the ion temperature T i ⫽0.03 eV did not agree with the measured P and A i dependence. The calculated v⬜i for T i ⫽30 eV showed a similar dependence on P in the pressure range of P⯝10⫺4 ⫺10⫺2 Torr and A i as the experimental observations, i.e., v⬜i decreases with P and A i . However, disagreements with the experiments occur at the lower pressure region ( P⯝10⫺5 ⫺10⫺4 Torr). Furthermore, the absolute calculated value of v⬜i at T i ⫽30 eV is more than two orders of magnitude larger than the measured v Di . It was shown that some instabilities such as the twostream ion cyclotron instability,13 the current convective instability,12,14 and the Simon–Hoh instability4 enhance the radial loss rates and give rise to anomalous diffusion or turbulent diffusion. Turbulent ion heating caused by instabilities such as the ion acoustic wave,15 the ion cyclotron drift wave,16,17 and the two-stream lower-hybrid instability18 has been studied. For the lower-hybrid heating case, even though the instability amplitude showed a discrete-spectrum, stochasticity was introduced because of the large growth rate of the instability.18 We postulate that the stochastic heating by the MSHI occurs in our experiments. In order to resolve this issue, time resolved ion temperature measurements are being contemplated. The instability onset measurements have also been conducted by applying the pulsed acceleration voltage V B . We have observed that onset is inversely proportional to P and becomes shorter for the larger I b . 6 Figure 2 shows the radial position dependence of the time variation of the electron
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Phys. Plasmas, Vol. 7, No. 5, May 2000
Growth and nonlinear evolution of the modified . . .
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saturation current I es . At the plasma center 关Fig. 2共a兲兴, increase in the dc level 共formation of the plasma due to ionization兲 is observed early in time. A fluctuation starts to oscillate roughly 2.5 s later. At r⫽0.5 cm, following a slow increase in the dc, a clear instability oscillation starts 共not shown兲. The amplitude of the oscillation reaches the saturated level after two cycles. As shown in Fig. 2共b兲, at r ⫽1.0 cm, neither a dc increase nor a fluctuation is observed until t⯝13 s. A clear oscillation starts at t⭓13 s and the fluctuation level is already saturated at the first cycle with n 1 /n 0 ⯝0.93. These results imply that at r⭓0.5 cm the electrons are pushed out by the fluctuation. We also find that the instability growth time growth is on the order of the instability period, i.e., I ⯝ R ; irrespective of the radial position. Figure 2共c兲 shows a plot of the dispersion relation obtained by applying Eqs. 共1兲 and 共2兲 to Poisson’s equation.1 We see that I can be the order of R , i.e., the instability is in the strongly driven regime as predicted by theory. IV. NONLINEAR EVOLUTION OF THE MSHI
As I b is increased the MSHI or M 1 mode grows and we observe a new low-frequency mode M 2 with a frequency f 2 , harmonic frequencies of f 1 and f 2 , and the various beat frequencies. In other words, the frequencies are linear combinations of f 1 and f 2 given by f ⫽m f 1 ⫹n f 2 (m,n⫽0,⫾1, ⫾2,...). In many cases, the spectral amplitude of the lowersideband with frequency f 1 ⫺ f 2 is larger than that of the upper-sideband with frequency f 1 ⫹ f 2 . Therefore, we represent the sideband mode M s with the lower-sideband component with frequency f s ⫽ f 1 ⫺ f 2 . We define the spectral amplitude of M 1 , M 2 , and M s modes as n 1 , n 2 , and n s , respectively. M 2 mode is an m⫽0 mode with n 2 that peaks at the plasma center and has a deep null at about the beam edge 关Fig. 1共b兲兴, together with a rapid radial phase change of 180 degrees 共not shown兲.6 M s is an m⫽1 mode with n s that peaks near the beam edge 关Fig. 1共d兲兴 like M 1 . It is found that6 f 2 is proportional to the square root of the electron density fluctuation level associated with M 1 which is ⬀ 冑˜n e . Furthermore, f 2 shows a reasonable correlation with the calculated value of the azimuthal (m⫽1) bounce frequency f b ⫽(1/2 )(e 1 /M r 2 ) 1/2 of the ions,5 where 1 is the fluctuating potential of the M 1 mode measured at r ⫽0.2– 0.5 cm by using an emissive probe 关Fig. 1共f兲兴. We also found that, while the dc plasma potential showed no dependence on the ion mass, 1 was found to be ⬀M . Since f b ⬀ 冑 1 /M , and f 2 ⯝ f b , f 2 was thus found to be independent of the ion mass. The nonlinear state of the MSHI can be controlled by changing either I b or P. Figure 3 shows I b dependence of the sequence of the nonlinear evolution of the MSHI. For I b ⬍1/ A, only M 1 mode is excited. As I b is increased M 2 mode appears together with M s mode. The ratio f 1 / f 2 decreases with I b . For I b ⬎36 A, strong frequency locking occurs at f 1 / f 2 ⫽3 and a third low-frequency mode M 3 appears with a frequency f 3 ( f 3 ⬍ f 2 and f 1 / f 3 ⫽18) together with the beat components m f 1 ⫹n f 2 ⫹p f 3 (m,n,p⫽⫾1, ⫾2,...). f 3 ( f 2 ⫺ f 3 ) increases 共decreases兲 with I b , and fre-
FIG. 3. I b dependence of 共a兲 frequencies f 1 , f 2 , f 1 ⫺ f 2 , f 3 , f 2 ⫺ f 3 , f 4 , and f 5 ; and 共b兲 frequency ratios f 1 / f 2 共closed circle兲, f 1 /( f 1 ⫺ f 2 ) 共open square兲, f 1 / f 3 共closed triangle兲, and f 1 / f 4 共open circle兲. Ar 1⫻10⫺5 Torr, r⫽0.4 cm, and z⫽2.5 cm from the electron gun.
quency locking of f 3 occurs at f 1 / f 3 ⫽6 or f 2 / f 3 ⫽2. At I b ⫽81 A, in addition to the frequency locked modes M 1 , M 2 , and M 3 , two new low-frequency modes M 4 and M 5 appear with frequencies f 4 ⫽ f 1 /12 and f 5 ⫽ f 1 /24 关 f 1 / f 5 is not shown in Fig. 3共b兲兴, respectively, which give frequencies f 5 ⫽ f 4 /2⫽ f 3 /4⫽ f 2 /8⫽ f 1 /24, i.e., 3⫻2 3 discrete frequency sub-harmonic components. M 4 and M 5 disappear at I b ⫽86 A, whereas M 1 , M 2 , and M 3 are still locked. M 3 disappears at I b ⫽91 A, and further increase in I b causes unlocking of M 1 and M 2 . At I b ⫽121 A, f 2 and f s jump toward f 1 /2, and are locked at f 1 / f 2 ⫽2 together with the appearance of unlocked M 3 at f 1 / f 3 ⯝4⫺5. M 3 is locked at I b ⫽213 A, and M 4 and M 5 appear with frequencies f 5 ⫽ f 4 /2⫽ f 3 /4⫽ f 2 /8⫽ f 1 /16. Note that, for I b ⫽106 ⫺152 A, f 1 decreases with I b or with f s , i.e., frequency pulling occurs. Frequency-locked M 2 disappears at I b ⫽358 A, and M 1 is the only mode that exists, i.e., remarkably we return to the original coherent state containing a single mode that is sitting atop an even smaller noise level. Instead of using I b if both I b and P are used to control the evolution of the MSHI, we see the onset of a similar sequence that ends quite differently. A typical example of this at f 1 / f 2 ⫽3 is shown in Fig. 4.5 Frequency locking occurs at f 1 / f 2 ⫽3 in Fig. 4共a兲. Decrease in P leads to the appearance of M 3 with a frequency f 3 and the beat components 关Fig. 4共b兲兴. With a slight decrease in P, f 3 migrates towards f 2 until f 3 locks with f 2 such that f 3 ⫽ f 2 /2⫽ f 1 /6 关Fig. 4共c兲兴. With a slight increase in I b causes the appearance of M 4 with a frequency f 4 关Fig. 4共e兲兴, migration of f 4 toward f 3 and frequency locking with its lower-sideband at f 3 /2 关Fig. 4共e兲兴. Up to five cascades of decays which give frequencies f 5 ⫽ f 4 /2⫽ f 3 /4⫽ f 2 /8⫽ f 1 /24 are observed 关Fig. 4共f兲兴. Any further increase in the control parameter 共P or I b )
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Phys. Plasmas, Vol. 7, No. 5, May 2000
FIG. 4. Observation of a cascade of sideband instabilities when f 1 / f 2 is locked at three. The vertical scale in 共a兲–共f兲 is log10 of spectral amplitude, in dBV. 0 dBV⫽1 V. Ar pressure is decreased from 共a兲 1.2⫻10⫺5 Torr to 共f兲 7.2⫻10⫺6 Torr. I b ⫽100– 170 A and r⫽0.5 cm.
leads to frequency unlocking of all the modes, which is shown in Fig. 5. An increase in P and I b causes a transition from a frequency locked state at f 4 ⫽ f 3 /2⫽ f 2 /4⫽ f 1 /12 关Fig. 5共a兲兴 to that with unlocked ( f 1 / f 2 ⫽3.12) components as shown in Fig. 5共b兲, where various peaks are no longer identifiable as linear combination of the modes, their harmonics and their beat frequencies. The phases of all the modes are also random at this stage. Because of the rather good correlation between frequency f 2 of the mode M 2 and the bounce frequency f b 关Fig. 1共f兲兴, we have postulated that when the M 1 mode attains a
FIG. 5. Observation of evolution of instabilities at r⫽0.4 cm. 共a兲 Frequencies are locked at f 4 ⫽ f 3 /2⫽ f 2 /4⫽ f 1 /12. Ar 7.7⫻10⫺6 Torr and I b ⫽116 A. 共b兲 f 1 and f 2 are unlocked ( f 1 / f 2 ⫽3.12). Ar 7.8⫻10⫺6 Torr and I b ⫽132 A. The vertical scale in 共a兲 and 共b兲 is log10 of spectral amplitude, in dBV.
Y. Sakawa and C. Joshi
FIG. 6. I b dependence of 共a兲 frequencies f 1 , f 2 , f 1 ⫺ f 2 , f 3 , and f 2 ⫺ f 3 ; 共b兲 frequency ratios f 1 / f 2 共closed circle兲, f 1 /( f 1 ⫺ f 2 ) 共open square兲, f 1 / f 3 共closed triangle兲, and f 1 /( f 1 ⫺ f 3 ) 共open circle兲; and 共c兲 amplitudes of the frequency components in 共a兲. In 共c兲 symbols are the same as in 共a兲. Ar 1 ⫻10⫺5 Torr, r⫽0.3 cm, and z⫽2.5 cm from the electron gun.
sufficiently large amplitude, it can trap a significant number of ions in the wave potential and may be driven modulationally unstable to sideband modes M s 19 with frequencies in the laboratory frame of f 1 ⫾ f b ⫽ f s . Simultaneously, the M 2 mode with a frequency f 2 ⫽ f 1 ⫺ f s and azimuthal mode number m⫽0 is excited in the plasma. As 1 increases, f 2 migrates towards f 1 ( f 2 ⫽ f b ⬀ 冑 1 ) and are mode-locked at f 2 ⫽m f 1 /n (m,n⫽1,2,...). This new periodic state might be further modulationally unstable to a low frequency mode M 3 with a frequency f 3 which will then migrate and mode-lock at f 3 ⫽m f 2 /n, and so on. Computer simulations are necessary to give insights into this cascade coupling route described above. We have described one scenario for the nonlinear evolution of the MSHI: Coupling to a sequence of sideband instabilities that arise because of trapped ions. However, there is another path that the nonlinear evolution of MSHI can take even though the experimental conditions are nearly the same as described below. Figure 6 shows a different sequence of wave–wave couplings as I b is varied. The only difference in the experimental condition is that the diameter of the endplate is now 1.2 cm instead of 5 cm previously. Evolution of the MSHI for I b ⬍38.9 A is similar to that shown in Fig. 3. However, at I b ⫽38.9 A, in addition to M 1 , M 2 , and M s modes ( f 1 / f 2 ⫽2.38), a new mode M 1/2 appears at a frequency
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Phys. Plasmas, Vol. 7, No. 5, May 2000
FIG. 7. A sequence of an energy exchange between M 1/2n mode and M 2 , M s modes. 共a兲 V B ⫽190 V and I b ⫽205 A, 共b兲 V B ⫽195 V and I b ⫽226 A, 共c兲 V B ⫽200 V and I b ⫽230 A, 共d兲 V B ⫽220 V and I b ⫽220 A. Ar 1⫻10⫺5 Torr and r⫽0.3 cm.
f 1 /2, i.e., M 1 mode undergoes a period doubling, which is clear in Fig. 6共b兲. At I b ⫽53.9 A, M 1 and M 2 are locked at f 1 / f 2 ⫽2 and merge to M 1/2 mode. At I b ⬎74 A, M 1 and M 2 are unlocked in frequency and f 2 is now larger than f 1 /2. However, M 1/2 still exists with a frequency f 1 /2. At I b ⬎300 A, M 2 and M s mode amplitudes become smaller and disappear together at I b ⫽410 A 关see Fig. 6共c兲兴, while the M 1/2 mode amplitude keeps increasing until I b ⫽410 A. At I b ⫽510 A, M 1/2 bifurcates into two modes 共not necessarily at a subharmonic frequency兲, and they disappear at I b ⫽600 A. In this sequence we have observed the excitation of M 1/2 mode when f 2 and f s are getting closer to f 1 /2 and frequencies are unlocked. There are two groups of the subharmonics of f 1 : One is M n⫹1 and M s n⫹1 modes whose frequencies are f n⫹1 and f s n⫹1 ⫽ f 1 ⫺ f n⫹1 , respectively (n⫽1,2,...). These are the modes which we have described earlier. The other is M 1/2n modes with frequencies m f 1/2n These modes appear as a result of period doubling sequence. We have observed an energy exchange between these two groups as will be shown below. Figure 7 shows a sequence of the energy exchange between M 2 , M s modes and M 1/2n modes. In Fig. 7共a兲 we find that the frequency components correspond to f 1 , f 2 , f s ⫽ f 1 ⫺ f 2 , 2 f s , 3 f s , f 1 ⫺2 f s , and f 1 ⫺3 f s . f 2 is slightly larger than 5 f 1 /7 and frequencies are unlocked. This causes the excitation of sidebands at each frequency component. In Fig. 7共b兲, frequency locking occurs at f 1 / f 2 ⫽5/7, and new frequency components appear at f 1 /2 and f 1 /2⫾ f s . Figure 7共c兲 shows the increase in the amplitude of f 1 /2 component and the appearance of f 1 /4 and 3 f 1 /4 components. Note that in this case, a narrow frequency locking at f 1 / f 2 ⫽32/23 occurs. In Fig. 7共d兲, we see that the amplitude of f 1 /2, f 1 /4, and 3 f 1 /4 components increase by 20.25, 15.81, and 31.26 dBV, respectively, together with the decrease in those of f 2 and f s components by ⫺9.65 and ⫺10.65 dBV, respectively. Frequencies are locked at f 1 / f 2 ⫽10/7. When f 1 /2, f 1 /4, and 3 f 1 /4 components appear or period doubling sequence oc-
Growth and nonlinear evolution of the modified . . .
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FIG. 8. Variation of 共a兲 amplitudes of the frequency components f 1 共closed circle兲, f 2 ⫽ f 1/2 共open square兲, f 3 ⫽ f 1/4 共closed circle兲, f 4 共open circle兲, and f 5 共closed square兲; and 共b兲 frequency ratios f 1 / f 2 ⫽2, f 1 / f 3 ⫽4, f 1 / f 4 , and f 1 / f 5 . In this sequence, current to the filament cathode is kept constant and V B is varied from 225 V (I b ⫽926 A) in Number 1 to 242 V (I b ⫽976 A) in Number 14. Ar 1⫻10⫺5 Torr and r⫽0.35 cm. Amplitude of f 1/2 component dependence of 共c兲 frequencies f 1/2 , f 1/4 , f 4 , and f 5 ; and 共d兲 amplitudes of f 1/4 , f 4 , and f 5 . Symbols in 共c兲 and 共d兲 are the same as in 共b兲.
curs 关Figs. 7共a兲–7共c兲兴, amplitude of f 1 component increases by 0.21 dBV and those of f 2 and f s components are unchanged within 0.04 dBV. On the other hand as seen in Figs. 7共c兲–7共d兲, the increase in the amplitude of f 1 /2, f 1 /4, and 3 f 1 /4 components seems to be sustained by the decrease in those of f 2 and f s components, since the amplitude of f 1 component increase only by 0.03 dBV. Therefore, we believe an energy exchange from f 2 and f s to n f 1 /4 components occurs. Figures 8共a兲 and 8共b兲 show the evolution of the instabilities when f 2 locks at f 1 / f 2 ⫽2 ( f 2 ⫽ f 1/2). In this sequence, current to the filament cathode is kept constant and the acceleration voltage V B is varied. When V B is varied from 225 V 共Number 1兲 to 242 V 共Number 14兲, I b increases from 926 to 976 A and f 1 decreases from 71.75 to 67.5 kHz. A new mode M 3 appears with a frequency f 3 exactly at quarter subhamonic of f 1 ( f 3 ⫽ f 1 /4⫽ f 1/4). Whereas, a lowfrequency M 4 mode appears at a frequency f 4 ⯝ f 1 /21.4. f 1 / f 4 tends to decrease by increasing V B , and when frequency locking occurs at f 1 / f 4 ⫽8, a new mode M 5 with a frequency f 5 ⯝ f 4 /2⫽ f 3 /4⫽ f 2 /8⫽ f 1 /16 appears. Therefore, both the period doubling and low-frequency excitation/mode locking sequences can simultaneously occur. Note that in this sequence, the amplitude of f 1 component is nearly unchanged, whereas, that of f 1/2 is slightly increased. In Figs. 8共c兲 and 8共d兲, frequencies and the amplitudes are replotted versus the amplitude of f 1/2 component, respectively. The excitation of the subhamonic modes can be seen in these figures, i.e., M 1/4 mode is excited when the amplitude of M 1/2 mode becomes large enough, and M 4 mode is excited when the amplitude of the M 1/4 mode becomes large. At present, there is incomplete physical understanding of these phenomena described above, i.e., the physical mechanism that leads to period doubling of M 1 mode and the mechanism that leads to the half-harmonic mode growing at
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the expense of M 2 and M s modes are not clear. We believe that the explanation for these observed effects will have to come from 3D 共three-dimensional兲-computer simulations. This is currently being explored. Finally, there are other paths the nonlinear evolution of the MSHI can follow particularly if the plasma length L p is made shorter. These will be discussed in a future publication. V. CONCLUSIONS
The growth and nonlinear evolution of the MSHI is observed in a weak electron beam-produced collisionless cylindrical plasma, in which electrons are strongly magnetized and the ions are essentially unmagnetized. Transient study of the MSHI reveals that the growth rate of the MSHI is the order of the instability frequency. One path the evolution of this instability takes is through a sequence of sideband instabilities, thought to be induced by trapped ions. A second path is a period doubling sequence. It is not known why the system chooses one path to nonlinear evolution over another when different control parameters 共beam current, beam voltage, and gas pressure兲 are varied. It is clear that this plasma is an excellent testbed for understanding how wave–wave and wave–particle interactions lead to a complex spectrum. The ultimate goal of this work is to trace the development of plasma turbulence in this model system from first principles. ACKNOWLEDGMENTS
We thank F. Wang, S. Wang, and K. Marsh for their help with the experiment, and Dr. P. K. Kaw, Dr. T. W.
Y. Sakawa and C. Joshi
Johnston, Dr. F. F. Chen, and Dr. J. M. Dawson for many stimulating discussions. Work was supported by the U.S. Department of Energy/National Science Foundation Plasma Science Initiative.
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